MAT 530 Topology, Geometry I, Fall 2009.
The final exam is on Wednesday, Dec 16, 2:154:45 in Physics P124.
 Instructor: Olga Plamenevskaya, office 3107 Math Tower,
email: olga@math.sunysb.edu
 Office hours: MF 10:40am11:40pm in P143
or by appointment.
 Grader: Alexandra Popa, office 3105 Math, email:
alexandra@math.sunysb.edu,
 Class meetings: Monday and Friday, 12:502:10pm, Physics P124.
References :
 James R. Munkres, Topology, 2nd edition.
Munkres's book is available in the campus bookstore (but can certainly be found for less money elsewhere).
It is the required
text. Parts of the homework will be assigned from it, and there will be required readings.
Other useful books:
Allen Hatcher, Algebraic Topology.
This book is available for free from Hatcher's webpage;
Chapter 1 that covers the fundamental group is here.
William S. Massey, Algebraic Topology: An Introduction, GTM
56. (The first few chapters are also contained in another book by Massey,
GTM 127). Massey has a nice treatment of classification of surfaces.
Homework: weekly assignments will be posted on
this page. Homework will constitute a significant part of your course grade.
Important: Please
write up your solutions neatly, be sure to put your name on them and staple all pages.
Illegible homework will not be graded. Late homework will not be accepted.
You are welcome to collaborate with others and even to consult books,
but your solutions should be written up in your own words,
and all your
collaborators and sources should be listed.
 Homework 1: Munkres §13: 1, 5 (basis part only), 8
and
five more questions (pdf), due Sept 11 .
 Homework 2: Munkres §16: 8 ; §17:
6, 7, 19, and
three more questions (pdf), due Sept 18.
Required Reading: please read §18.
 Homework 3:
pdf, due Sept 25.
Required Reading: please read §25.
 Homework 4:
pdf, due Oct 2.
 Homework 5:
pdf, due Oct 9.
 Homework 6:
pdf, due Oct 16. 10/13: mistake in Problem 1 corrected.
 Homework 7:
pdf, due Oct 23.
 Homework 8:
pdf, due Nov 6.
 Homework 9:
pdf, due Nov 13.
 Homework 10:
pdf, due Nov 20.
 Homework 11:
pdf, due Dec 4. Please start early, it's a big one!
 Homework 12:
pdf, due Dec 11. The very last one!
Some solutions (or at least some hints)
pdf,
 Syllabus: we will follow the basic outline from
the graduate core course requirements (see below), not necessarily in the same order, with
additional topics
as time permits.

Basic point set topology
 Metric Spaces
 Topological spaces and continuous maps
 Comparison of topologies
 Separation axioms and limits
 Countability axioms, the Urysohn metrization theorem
 Compactness and paracompactness, the Tychonoff theorem
 Connectedness
 Product spaces
 Function spaces and their topologies, Ascoli's theorem
 Introduction to algebraic topology
 Fundamental group
 Fundamental group of
S^{n};
examples of fundamental groups of surfaces
 Seifertvan Kampen theorem
 Classification of covering spaces, universal covering spaces; examples
 Homotopy; essential and inessential maps
Students with Disabilities: If you have a physical,
psychological, medical, or learning disability that may impact on your
ability to carry out assigned course work, you are strongly urged to
contact the staff in the Disabled Student Services (DSS) office: Room
133 in the Humanities Building; 6326748v/TDD. The DSS office will
review your concerns and determine, with you, what accommodations are
necessary and appropriate. A written DSS recommendation should be
brought to your lecturer who will make a decision on what special
arrangements will be made. All information and documentation of
disability is confidential. Arrangements should be made early in the
semester so that your needs can be accommodated.